212 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
as they are, rather than to expand them in powers of <?, and S. In like manner the 
terms of the second order are 
i+«+V\" a ii + i 
1+e + A 2A (l+e + d) 
the terms of the third order are 
1 + e + 3 
5 
;{£e a +io,(B',+ 3)} 
1 + e + d 
c 3^a 3 + ^Y +e + ^)5 i2 c 3+ a c 2 (B'i + 2) +fci (B'j + 1) (B'j + 5) +fciB' 2 } (c> ,! — <V) 
+ 
- C x e ($ — $ A )- — 
(l+e + 8) 
-c x B\ (S + 2S x )(S-S A y, 
(l +e + S)' 
and the terms of the fourth order are 
y-j——r i C A 4 + ^ + {i" c 4 +f c 3 (3Bh + 5) +fc 2 (B'i +1) (B'x + 3) + fc 2 B' 2 
+ 4 Ci (B'j +1) 2 (B'i + 7) +-§Ci (B'i + 3) B'y+fciB'g} (d 4 — S A ) 
+ ^Y~ t +3)' ; + ^ Cl + } 6 + 2d A ) ~4) 2 
— ( 1+6+ ^ 1 ^ C2 ® /l + ^ Cl ® /]1 (SBh + ll) + ‘ 2 6 5 CiB' 2 } (d a +2<5 A ^+3^ A 3 ) (8— fl A ) 2 . 
38. Formula for Z .—We write 71 for Z —Z 2 , and seek first a formula for 71 along the 
locus x 0 = 0. The value of 71 along this locus is given by the equation 
Z'=r P<2o-+f?cfa, 
J(R 2 . S 2 )0(7 cu 
where the integral is taken along the locus dZ/dcr — 0, so that 
71 = f t^ds = So f(T 2 +c 1 ^+c/+...)(B , 1 -l + 2BV+3BV 2 +...)^. 
j St ds J o 
Thus the value of 71 along the locus can be expanded in powers of S in the form 
71 = d 1 $+d 2 8 2 +d s S»+..., 
where 
d 1 = S a T J (B , 1 -l) > 
C ? 2 = S 2 {T 2 B' 2+ i Cl (B , a -l)} 3 
d, = S 3 {T 3 B' a + |c,B' 2+ ic 2 (B' 1 -l)}, 
d, = ^{T.B'. + fcB^ + ic^+^fB'.-l)}, 
d- 0 — S 2 {T 2 B' 5 +| Cl B' 4 +lc 2 B' 3 +|c 3 B' 2 + \ Ci (B i-1)}, 
