384 PROCEEDINGS OP THE AMERICAN ACADEMY. 



and shall itself pass through those points. Then the change in the 

 integral due to changes in the co-ordinates Spi, 8 p.2, 8 p^, is, 



=/* (rO '"' + ' (x) '''>•- + ' (CO ''"' + S '"''^ 



P P 



+ -^ d8p2 + y^ '-/Spg. 

 The last three terms may be integrated hy parts, giving 



B B B 



and the integrated part vanishing at the limits, 



' ' =/' © '"' + ' © ■'"' + ' (t) ■'"= - '"^ ' 



— 8p^d[ -j^ ] — 8psd ij^ 



Performing the operations denoted by 8 and d, and collecting the 

 terms which do not cancel, 





+ ( 8ps dp, -8prdps ] J ^ ( V ) -rz ( ^J I 



8 Pa \ /'i J 8pi 



'Pi\ 8 (P. 



) \^. \ 



-\- [ 8 pi d po — 8 pn d pi 



{ 8 Pi d pn — 8 p.2 d Pi I < 





Now the changes 8 p„ d p^, \n the co-ordinates correspond to dis- 

 tances — ^% y^, measured along the co-ordinate lines, and the de- 

 terminant of these distances, 



— — (8p2 d ps — 8ps dp2), 



«2 ^S 



