480 Tlie Rev. J. Challis on the Principles of the 



spect to space is in tiie direction of the motion, it follows that 

 the transformed equation and its integral containing arbitrary 

 functions must be immediately applied to the parts of the 

 fluid disturbed in a given and arbitrary manner, and where, 

 consequently, the direction of the motion is known. The ve- 

 locity, and direction of the velocity, at other parts are to be 

 inferred from the laws of propagation of the motion and 

 variation of the velocity which the integration reveals. 



Admitting the extension of the foregoing considerations to 

 a compressible fluid, the following is the manner in which the 

 problem of the resistance to a vibrating sphere requires, on 

 these principles, to be treated. The gtMieral equation in rect- 

 angular coordinates, viz. 



Tt^ - " • \d^' -^dj^ d?y 



is first to be transformed into polar coordinates, having the 

 centre of the vibrating sphere for the pole. The polar co- 

 ordinates being r the distance of the point x y ^ from the cen- 

 tre of the sphere, 6 the angle which /• makes with the straight 

 line in which the centre is moving, and >; the angle which the 

 plane of these two lines makes with a vertical plane, regard 

 also being had to the motion of the sphere, the translbrmation, 

 effected in the manner adopted by Poisson in Art. 2 of his 

 Memoir in the Connaissance des Terns for 183't, leads to the 

 equation 



1. ^ 1 7^ -^ 1 "• I Sin t^ —5-75— I 



d' . r ^ _ 4 ^ d- . )■ 4> 1 \ dO J 



T7' ^' ' L dt'' "^ r' sin ' Je 



1 d- . r (p 



/•- sin'- 6 ' d >)'- 



which is independent of the motion of the sphere. If now 

 the reasoning be just by which I have argued, that into this 

 equation must be introduced the condition, that the variation 

 with respect to space be in the direction in which the velocity 

 is impressed, it will plainly reduce itself to 



d'- . r <p _ „ d- . r f 



~TF ~ "' ' 'ch^ ' 

 in which, from what has been already said, ^ may contain as 

 a factor an arbitrary function of 6 and ij. 'I'his is the equa- 

 tion 1 have employed in the solution I have given of the pro- 

 blem in the Number of this Journal for December 1840, and 

 by it I have been conducted to the value 2 for the coefficient 



