232 The Rev. J. Challis on a new Method of Investigating 



Hence by substituting these values of w, v 3 and to in the equa- 

 tions (a), (b), (<?), given in page 64 of the July Number, viz. 



dV , . du _ rfP , , dv _ rfP , . rfw „ 



« # a/ dy dt dz dt 



and neglecting quantities such as —7- X ~, which will be of 



the order of the square of the velocity, it will be found that 

 dP k_ d$ _ 

 Tt + N'dT~°' 

 Hence from the equation 



dP , du dv dw _ 

 Tt^Jx + Ty* d~z~°' 

 it follows (since k a 2 = 1 ) that 



dt*~ a ' U* 2 + dy* + dz*j 



_a* fdN ±±,dW ^>>f[N dj\ 

 N * \dx ' d x dy ' dy dz ' d z J 



Let us now transform this equation into one of polar coor- 

 dinates r, 0, >), the angle being that which the radius vector 

 r makes with the axis of z, and ij that which the plane of these 

 two lines makes with a vertical plane. And, guided by the 



foregoing result, let us assume the factor N to be , or 



z — c 



-.. By the usual rules of transformation I find that 



COS0 J 



sin20^ r ^ 



dP ~ a \dr* + r 2 sin20d0 + r 2 sin 2 d f J ' 



an equation differing from Poisson's in having sin 2 in the 

 place of sin in the second term of the right-hand side, and 

 also in the signification of <p. This equation, applied to the 

 instance of the oscillating sphere, remains the same (small 

 quantities being neglected) if the origin of the polar coordi- 

 nates be the centre of the sphere. The resolved parts of the 

 velocity in the directions of the three rectangular axes being 



d $ . d $ d <p 



-j- cos 0, -7— cos 0, — 1 cos 0, 



dx dy dz 



the velocity in the direction of r will be -~ cos 0. Now the 



dr 



