358 



Mr MURPHY'S THIRD MEMOIR ON THE 



an ordinate drawn near the origin at any small distance not varying 



with the parameter Ji, and since -r- has the same sign in the interval 



from B to H', H or A, it is evident that the portion of the curve 

 BE' tends to coincide with the axis BH', the curve therefore which 

 represents V„ coincides with AB, except infinitely near the origin 

 A, when it suddenly mounts to an infinite height. 



Since the general function V„ is reciprocal to t", it follows that 

 fi Vaf = 0, except when w = 0, and then the definite integral is unity ; 

 hence if f{t) be any function containing only the positive and integer 

 powers of t, the transient function Vo possesses the remarkable pro- 

 perty expressed by the equation [tV'o ■/{f)=J^{Q). 



Fig. 2. Let 2a = AB, equal the 

 length of the axis in a solid of revo- 

 lution, the surface of which is covered 

 with an indefinitely thin stratum of 

 fluid, let any abscissa ON measured 

 from the centre O be put equal to 

 a (1 - 2#), the limits of t will evidently 

 be O and 1 . 



Let the law of density or accumulation at any point P of a section 

 perpendicular to the axis be expressed by the transient function \V^, 

 X being constant, and let the total action of the fluid on any point Q 

 in the axis be required, the law of force being capable of expansion 

 according to the positive and integer powers of t. 



Put PA'' = y, then the whole quantity E of fluid is manifestly 



ds 

 equal to ^Xtt ft T^^y -r, , s representing the arc AP. 



ds 

 Now it is easily seen that the value of y-yr at the point A where 



y vanishes is iaR, R being the radius of curvature at that point, and 

 by the nature of V^ this quantity is the value of the above integral, 

 or E = SXttuR. 



