Changes

Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'', ''v''].  The critical step of this procedure uses the facts that (0, ''x'') = 0 + ''x'' = ''x'' and (1, ''x'') = 1 + ''x'' = (''x'') for any boolean variable ''x''.

Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'', ''v''].  The critical step of this procedure uses the facts that (0, ''x'') = 0 + ''x'' = ''x'' and (1, ''x'') = 1 + ''x'' = (''x'') for any boolean variable ''x''.

<pre>

<font face="courier new">

Table 38.  Computation of EJ (Method 1)

o-------------------------------------------------------------------------------o

|+ Table 38.  Computation of E''J'' (Method 1)

|                                                                               |

| EJ = J<u + du, v + dv>                                                      |

{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

|                                                                               |

| width="8%"  | E''J''

|     =  (u, du)(v, dv)                                                         |

| width="4%" | =

|                                                                               |

| width="44%" | ''J''‹''u'' + d''u'', ''v'' + d''v''›

|     =   u v J<1 + du, 1 + dv> +                                            |

|                                                                               |

|         u (v) J<1 + du, 0 + dv> +                                            |

|                                                                               |

|-

|       (u) v J<0 + du, 1 + dv> +                                            |

| width="8%"  | &nbsp;

|                                                                               |

| width="4%" | =

|       (u)(v) J<0 + du, 0 + dv>                                              |

| width="44%" | (''u'', d''u'')(''v'', d''v'')

|                                                                               |

| width="44%" | &nbsp;

|     =   u v J<(du), (dv)> +                                                |

|-

|                                                                               |

| &nbsp;

|         u (v) J<(du),  dv >  +                                                |

|-

|                                                                               |

| width="8%"  | &nbsp;

|       (u) v J< du , (dv)> +                                                |

|                                                                               |

| width="44%" | &nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;''J''‹1 + d''u'', 1 + d''v''›

|       (u)(v) J< du , dv >                                                  |

|                                                                               |

o-------------------------------------------------------------------------------o

| width="8%" | &nbsp;

|                                                                               |

| width="4%"  | &nbsp;

| EJ  =   u v (du)(dv)                                                         |

| width="44%" | &nbsp;''u''&nbsp;(''v'')&nbsp;''J''‹1 + d''u'', 0 + d''v''›

|                       +   u (v)(du) dv                                      |

|                                           + (u) v du (dv)                   |

|                                                             + (u)(v) du  dv |

| width="8%" | &nbsp;

|                                                                               |

| width="4%"  | &nbsp;

| width="44%" | (''u'')&nbsp;''v''&nbsp;&nbsp;''J''‹0 + d''u'', 1 + d''v''›

</pre>

| width="8%" | &nbsp;

| width="4%"  | &nbsp;

| width="44%" | (''u'')(''v'')&nbsp;''J''‹0 + d''u'', 0 + d''v''›

| width="8%"  | &nbsp;

| width="4%"  | =

| width="44%" | &nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;''J''‹(d''u''),&nbsp;(d''v'')›

| width="8%" | &nbsp;

| width="4%"  | &nbsp;

| width="44%" | &nbsp;''u''&nbsp;(''v'')&nbsp;''J''‹(d''u''),&nbsp;&nbsp;d''v''&nbsp;›

| width="8%" | &nbsp;

| width="4%"  | &nbsp;

| width="44%" | (''u'')&nbsp;''v''&nbsp;&nbsp;''J''‹&nbsp;d''u''&nbsp;, (d''v'')›

| width="8%" | &nbsp;

| width="4%"  | &nbsp;

| width="44%" | (''u'')(''v'')&nbsp;''J''‹&nbsp;d''u''&nbsp;,&nbsp;&nbsp;d''v''&nbsp;›

|}

|-

|

{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

| width="8%"  | E''J''

| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')

| width="23%" | &nbsp;

| width="23%" | + ''u'' (''v'')(d''u'') d''v''

| width="23%" | &nbsp;

| width="23%" | &nbsp;

| width="23%" | + (''u'') ''v'' d''u'' (d''v'')

| width="8%"  | &nbsp;

| width="23%" | &nbsp;

| width="23%" | + (''u'')(''v'') d''u'' d''v''

|}

|}

</font><br>

Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree.  Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring.  Consequential properties of exclusive disjunction then carry us on to the concluding line.

Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree.  Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring.  Consequential properties of exclusive disjunction then carry us on to the concluding line.