PEOF. H. M. MACDONALD ON THE EFFECT PEODUCED BY 
31fi 
Now 
n 
IK (Yt—rj—r) 
B 
RiR 
dS= - 
)iK(Yt-ri—r) 
RiR 
f/cr, 
where do- is the projection of the element of area dS on a plane perpendicular to the 
straight line OP; to evaluate this integral let the axes of reference be OP the axis 
of the perpendicular to OP in the plane of incidence the axis of x, and the straight 
line perpendicular to these two the axis of ?/. Let t), I, now denote the co-ordinates 
of a point on the surface, and |i, Oi, the co-ordinates of the point of contact R of 
the tangent from O to the surface in the plane of incidence; the co-ordinates of the 
point P are 0, 0, Ri-fR, then 
that is, the principal part of this integral is given by 
f CO I* GO ^ ^ 
— CO J — 00 ^ ^1 ^ 
It has already been proved that 
_2\R Ri/_ 
-t/<(Ki + R) . 
« CO ^ GO 
— jj. 
•J — 00*' — 00 
to obtain the value of the second integral 
= e+v^+ (Ri+R-cn 
and observing that ^i, correspond to the lower limit of the integral 
n = + &c., 
V — Ri -f R—^1-1- 
1 
+ &c.. 
therefore 
2{B, + B-C,) 
je—= e—jg-f ; 
hence, unless the radius of curvature of the projection of the curve of contact of the 
tangent cone witli the surface on the plane j^erpendicular to OP is a small quantity 
of the order of the principal part of the above integral is given l^y 
K)[" r 
Li, J - cc 
that is, the principal part of 
