66 Royal Society :— 
where wu, is the solution of 
; d\? d\? 
a 6 a 6 
Wafer tan) +F(e-0¥5 tan) F ‘ 
and the operation w,@n—) ... @)@, is easily seen to be equivalent to 
namely 
(sin 0)~”{ sin 6 bl sin 0) ee 
(This result is compared with that obtained in a different way by Pro- 
fessor Boole (Cambridge and Dublin Journal, vol. i. p. 18), to which 
it bears a general resemblance, but the author has not succeeded at 
present in reducing the one form to the other.) 
In the case in which u,, does not contain o, we have 
uy=C,+C, log tan >. 
The general expression for a “ Laplace’s coefficient’ of the nth order, 
not containing ¢, is therefore (sin a)-"(sin 6 hee ) .C; and if 
this be called v, when C=1, the development of (1 —2rcos 0+7°) ~* is 
9 
= 
Vy tyr +%, +... +n + 
7 ve 
1.2 UoQL ie dota 
and it is shown that the coefficient of i, in the development 
aad 
of (l—2rcos6+7") ? is 
; Bhp Oey Ne 
(sin 0)~"~* (sin 0 7 sin 0) (sin 6)'. 
With respect to the development of 
¢@ — 2r(cos 8 cos 6! + sin@sin@' cos) + r*) = 
it is shown that the coefficient of 7” cos ig may be put in either of the 
two forms, 
2 lat —27 2. pAl\—rorol® OV feud \' 
1 .2.:. (@—i).1.2.0 (apa) 9) (ain) "OO" | tang ( 5 
or 
2.17.3... (2i—1)? Svat? SPINY gee 52) ten eS Se 
ie... Leer eo ee 
where © represents the operation sin oF sin 6, and the factor 2 is in 
each case to be omitted when i=0. (This coefficient is a solution of 
the equation 
{ (sin 05) +n(n +1)(sin oye} u=0, 
ee 
