210 
and 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
dx 0 
0cr 
10 h 
-M£) 
dxg 
du 
10 h 
da 
du 
<T 1 — 
da- 
da 
where dajdu is to be found from the equation of the locus. 
Thus we may write 
dt 
0(7 
10 
W 1 
acr 11 
da 
du 0£ _ , 0 Ao- 0 10 1 
7 -7 _ ~ ’ 
i-d 
du 
aa 11 /cto-y 
\du J 
\du J 
Now let ( rs') be any point P, and let lines PA, PC parallel to the axes of s and r 
meet the locus in A, C, as in fig. 3 in Article 22. Then, since t satisfies the same differen¬ 
tial equation as Z, the integral 
fv(|*+5-)<fe + « 
j \as a/ 
dr 
taken round the contour formed by the arc AC and the lines CP, PA vanishes, and 
therefore we have the equation 
or, on putting 
the equation 
But we have, along the locus, 
0C = dd_ 
ds da 
dd 
du 
1 0 1 
aa 11 da 
du 
5//<t 0 10 du 
a a 11 ds 
and, by the theory of Article 25, the expression last written is the same as the value of 
Ttdtjds along the locus, or we have 
At' 
— = 1 ( c i + 2 c 2 S + 3c 3 0- +...). 
os 
Also along the locus we have 
d — Cl S + c 2 S 2 + c 3 S 3 + ... . 
