i8 
Mr. Ivory on the Method of computing 
viz. 
PC [ r — pf 1 • p a • v ‘ • . dm' 
|r*-2r P .y+ P *J*±i 
when it is extended to the whole surface of the sphere, and in 
the particular circumstance of r = p, or r — p = o. We must 
begin with transforming the formula to be integrated. The 
arcs 9 and 9' are the two sides of a triangle formed on the 
surface of a sphere; the angle contained by those sides is 
tut * — "zr ; and the third side of the same triangle is no other 
than the arc whose cosine has been denoted by y : let ® de- 
note the angle opposite to the side 9' whose cosine is p\ then 
if we suppose 9' and to vary, it has already been proved 
that the correspondent fluxion of the surface of the sphere 
will be = p 2 . dp . d'ur ’ ; but if we make y and <p vary, the same 
fluxion will be = P * . dy . d<p : therefore 
(r — pf " 1 . p 4, . v' . dpt . dm (r — pf 1 . p z . v' . dy . dtp 
| r'-zrp.y+fyjj ~ 
and as this is true for every element of the spherical surface, 
the fluents will likewise be equal when they are extended to 
the whole surface of the sphere. To complete the transforma- 
tion we must next convert o' into a function of y and <p ; after 
which the integration with regard to <p will be independent of 
the denominator in which y only is contained. Suppose o' to 
be actually transformed as here mentioned, then 
ff 
(r-p) 
i — i 
2rp . y+p 2 j 
. dm' 
1± 
2 
(Y 1 • p 2 • dy .fv . dip m 
|^_2rp. 7+ p 2 | / ±l 
the sign of integration in the numerator being understood to 
affect the variable <p only. 
