810 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
and electric forces cease to represent them is determined by the condition that the 
point Q related to the point P in the manner already defined is so close to the curve 
of contact of the tangent cone from the point O to the surface of the obstacle with 
the surface that the quantity (k-cP/Ri + is a quantity of the order of unity or 
of a higher order. 
When the point P is sufficiently distant from the surface of the tangent cone from 
the point 0 to the surface of the obstacle, the limits of the integral may be taken to 
be infinite, and the order of the difference between the comnonents of the actual 
magnetic and electric forces due to the obstacle and the components of the magnetic 
and electric forces due to tlie assumed electric current distribution may be obtained 
as follows. In the exponent of the terms up to and including those of the 
fourth degree in ^ and r) must be retained, and in the other factors of the integrands 
the terms up to and including the terms of the second degree in £ and rj must be 
]’etained ; this requires that in the equation to the surface 
C = l(AP+2Hf77 + Br/) + . 
terms uj) to and including the terms of tlie fourth degree in ^ and r) are retained, and 
the resultine' intea;rals are now of the form 
I {go + gi+9-2 + m + KC/,) e (I^ dy ], 
where g„ denotes a homogeneous function of rj of degree 7 l. The terms of odd 
degree in r) contribute zero to the result and the integral is equal to 
(do + 9-2 d$drj. 
Now, if the conducting surface were the infinite plane coinciding with the tangent 
plane at the point Q to the surface of the obstacle, the corresponding integral would 
be 
(do + 9'2 + xg'i) 
where tliis integral is equal to the value of the preceding integral when the quantities 
A, B, H, &c., in the equation to the surface are made zero. Further, for the surface 
C = 0, the condition that the tangential electric force due to the assumed electric 
current distribution vanishes at the surface is accurately satisfied, and therefore the 
principal j)art of the electric force tangential to the surface of the obstacle at the 
point Q is tlie value at Q of the principal part of an integral of the form 
j[L(.9'o-t.q2 + «’.b4) e '^""‘^^'^"^^''^^^ — {90 + 9'2 + Kd'i)^ c 
