158 Proceedings of the Royal Society of Edinburgh. [Sess. 
14, For the infinitesimal variations of a, 
we have 
a{x^, . , 
. . ) = a{x^, . . . 
) + 2e{a(ri, . 
, . — ) 
4- li{x ^ , 
• ■ • )iji(»i, ■ ■ ■ 
) 
+ g{x^, 
• ■ • )Si(*i > • • ■ 
) 
4" 1 (x^ , 
• • • )®i(*i, • • ■ 
)}. 
a(^r/4-af, . . 
. . ) = a(xj4-a^, 
. . . )-}-2e{a{x^ 
+ a, , . . . 
+ Oi, 
j • • • ) 
+ g{x^ + a^, . . . + . . . ) 
+ l (x^ + + ttj , . . . )}. 
We expand the various terms in the latter equation in powers and 
products of the quantities a and a, up to the second order inclusive. The 
terms independent of a and a balance, owing to the former equation. 
We then substitute the values of the quantities a in terms of the quan- 
tities a, retaining only the first power of e ; and we compare the coefficients 
of the various powers and products of the quantities a. The following are 
the results : — 
a/ = ^ -t- 
4- + 2hr]^r 
+ ^9r^i + 
+ '2ire^+a^e^ + 2id^r], 
for r = 1, 2, 3, 4 ; and 
^rs ~ ^rs ~ ^is^r ^\r^s ”1“ ^±^rs “1“ "f” 
4 ^2s9f ^2f'9s ”t 
^9rs^l + ^ss^r + ^sr^s + ^z^rs + ^9s^vr ^9 tQ\s + ‘^9^vrs 
+ 2/i-s^j^ -f ci/^sO^ 4- CL^yOg 4- ci^Ors + 4- 4- 2/dj^s] , 
for r, s= 1, 2, 3, 4 in all combinations. 
Proceeding similarly with the rest of the coefficients of the quadratic 
differential form, we obtain the results : 
—i>r = €\2h^^2 + ^1^2 2/i^2r 
+ 2/ + 2/ ^2r 
4- 2711^9^ 4- 4- 2mB^r ] ) 
brs ~ bfs ~ ^[2^^rs^2 b^s^r b-^j-^s "t ^i^rs ^bg^2r 2h^^2s 2Jl^2rs 
4- 2b^g7^2 + b2s7]r "t b2r9s b^^s 4* 2bg7j2r + 26^7^25 “t 2b7^2rs 
+ 2^/ rX>2 ^3s^r b^j.(^s 4- ^3^rs + 2fJ^2r 2fj.^2s 2f^2rs 
4- 2??lyg02 + ^4s^r + ^4r^s "I" ^4^rs + 2771^27 + 2m^02S + 2?W^£rs] ? 
Cr — Cy = e[2p'y^3 4- 4- 2g^^^ 
+ 2/ f.7j^ 4- c{gr + 2fy]^<r 
® 4- 2rj.^3 4* 4“ 2r^g^ 
4- 2nfi^ 4- c^Bf 4- 2nB^r] , 
