AND ENVELOPES OF HOMOLOGY. 



Forming the Jacobian, 



599 



du 

 df* 



du 

 dg' 

 du 

 dh ' 



dv 



V' 



dv 

 dg ' 

 dv 

 dh' 



dw 

 df 



dw 



dg 



dw 



dh 



= J 



(53), 



we have, 



(«/ + % + yh) x 



if, 



mg , 



nh , 



That is,- 



(«/ + 



a , @h + /(/ = J 



/3 , 7/ + «A 



7 f *g + Pf 



(54), 



(3g + 7 h).\ la ^ + l ^f - ^f - l 7«V + mfygh + mff - \ 



I ma 2 g 2 '- mafifg + nyahf + na 2 h 2 - nj3 2 h 2 - nfygh J " ( J ' 



Multiplying the part within brackets by the coefficient af, and differentiating 

 with respect to/, we have 



2lct 2 (3fg + 3Za/3 2 / 2 - 3la 7 2 f 2 - 2l 7 x 2 hf + mafygh + mfag 2 - 

 ma^g 2 - 2ma?(3fg + 2n 7 a 2 hf + na"h 2 - naj3'h 2 - nafygk 



(56). 



Differentiate, with respect to f, the part within the brackets of (55) in which / 

 appears, and multiply by (fig + yh), and we have 



la/3 2 g 2 + IW'fg - 21j3 7 2 fg - lafygh - ma/Pg* + na&ygh + 

 lufygh + 2l/3 2 7 hf - 2lfhf - l 7 2 ah 2 - mafygh + n 7 2 ah 2 



Hence 



(56) + (57) 



dJ 



dJ_ 

 df 



(57). 

 (58). 



Now, in this sum, we find in ji — 



Coefficient of/ 2 = 3Z«(/3- - f) . 



g* == a {(l - m)(3 2 + m(f - a 2 )} 



h 2 = «{(n - l) 7 2 + n{a 2 ~ p 2 )} 

 gh = 



hf = 2 7 {(n - l)a> + l((3 2 - 7 2 )} 



fg = 2<3{(/ - m )«? + Z(/3 2 - 7^)} j 



(59). 



And symmetrically for ^— and jr. 

 Now, it is well known that 



df V ' dg ~ ' dh ~ 



(60). 



VOL. XXIV. PART III. 



7z 



