188 
MR. ARTHUR SCHUSTER ON THE 
A further useful transformation is derived from the equations 
(2n+1) cr cos 6Q/ = cr (n—cr+ 1) QVtn + o- (n + cr) QVi, 
(2n+l) sin 0 = n.(n-a+ 1) Q <r „+i-(rt+ 1) (w + cr) Q%_i* 
If we subtract and add these equations, they reduce to 
and 
cr cos 
cr cos 
«Q.’~ sin e^f = sin 0Q»' +1 
6Q,” + sin 0 Asi = (n+<r) (n—<r+ 1) sin 6QC 1 . . . 
cl 9 
• (I*), 
• (L 2 ). 
We shall require to find the effect of the operation 
~ cos X ( j + cos X ~ sin 2 9 '! n ) Q/ cos (crX — a) . . . . (19). 
,c/X c/X c/9 do) ■ 
We omit, for the sake of shortness, temporarily the constant a, and divide the 
operation into two parts, the first being 
cos X ( -j— 0 + sin 9 — sin 9 ~ ) Q/ cos crX. 
c/X" cl9 
d9 
From the fundamental equation relating to tesseral harmonics this is equal to 
—. (n+ 1) sin 2 9Q," [cos (o-+1) X+ cos (cr— l) X] .... (20). 
The remaining part of the operation is 
c t sin X sin crXQ/ + sin 9 cos 9 cos X cos crX ( ! Q„ 
d9 
_ l 
dQn 
\ cos (o-+ 1) X (sin 9 cos 9 c/Q/—crQ/) + |- cos (cr — 1) X (sin 9 cos 9 y| L + crQ/) • (21). 
If (20) and (21) are now added, and K x and K 2 are applied, we find the result of 
the operation to be, restoring a, 
^ T {(»-l) ( n + 1) Qn-i a+1 ~n .n + 2 . Q„ +1 <r+1 ] cos {(cr+l)X-aj 
sin 9 
2. 2n + 
-{n(n + 2)(n — cr + 2)(n — cr+ l)Q )i+ i <r 1 — (n—l)(n+ l)(n + cr)(?i + a- l)Q„- 1 ,r 
cos {(cr— 1) X —a | 
■ ( 22 ). 
We may note that each of the equations used, and therefore the final results, 
remains true for cr = 0, if we define 
n. n +1. Q n 1 — — Q 7 ,, 
