184 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
In this expression we put 
b(T„ — r — (■g'O'o + s) — (?" + s) = 2" (< t 0 + rr — u) —rr, 
and find that the expression is the same as 
where 
0i i (T — u ) 
i 
Tcrs 
rf — -r(<T 0 + o- — u)° { 126rr 0 4 — 21 Oaf (<r 0 + fr — u) + 135(T 0 2 ((T 0 + cr — ll ) 2 
— -}~<r 0 (rr 0 + nr — uf + (<r 0 + a — uf} • 
The remaining terms in the expression for Z may be treated in the same way, and we 
obtain finally, as the expression for Z in the first middle wave, 
y _ ( C + h ) o-Q fl _8 Y J 01 {(T — u ) I li_ /1 _3 V f f t {g-u) \ 
9 Cl \<j cm 1 (T j <1 \cr cay { rr j 
_ H(T|, A _0_ \ 4 [ 0 ! (rr + u )\ _ H / 1_ jf \ 4 I fi (o- + m) [ 
9 a \ rr 3oy' 1 (T J « \(T Ocr/ I. rr j 
in which the expression denoted by 0i has been written down, and f, is given by the 
formula 
fi(o--ri) = - — 9 (o-o + cr-u)" (o- u —<T + u)"- 
It may be observed that the differential coefficient of the function <p x is given by 
the equation 
0 i (1) (rr —= ~ Jf'y-frr —w) 4 (rr„ — rr + U ) 4 . 
Zi 
Although the actual calculation of Z is rather long, it is comparatively easy to verify 
that the form obtained satisfies the conditions by which Z was determined. 
16 . Transformation of the Formula. —The form taken by Z in the first middle wave 
is 
I 1 ? Y [3>i (<r + u) +'k 1 (rr-u) l 
Vd~J l 7 1’ 
where 
'hi (rr + U ) = — 0! (<r + ll) — — fl (rr + u,), 
y ct w 
Ti ((T — u) = ^ ^ 0-1 1 01 (rr —w) + - f, (rr—a), 
so that dy and 4y are rational integral functions of the tenth degree. Now it is impor¬ 
tant to observe that when 1 8, 
rr errj {_ rr J 
