292 Lord Eayleigh on the 



In Laplace's notation the second member of (9). multiplied 

 by 27T, is represented by H. 



As Laplace has shown, the values for K and T may also be 

 expressed in terms of the function <f>, with which we started. 

 Integrating by parts, we get by means of (1) and (2), 



S+{z) dz=z*(z) +i; 3 II(c) +i \z*$(z) dz, 

 \zf(z) dz = iz?+(z) +izm(z) + t$*i(z) dz. 



In all cases to which it is necessary to have regard the inte- 

 grated terms vanish at both limits, and we may write 



\'iiz)dz=i \'^{z)dz, \%ir(z)dz = i Vz^{z)dz; (11) 



so that 



K =^f X r^)^. T : =l\^<f>(z)dz. (12) 



6 Jo °J© 



A few examples of these formulae will promote an intelli- 

 gent comprehension of the subject. One of the simplest 

 suppositions open to us is that 



#/)=*-* (13) 



From this we obtain 



n(*)=0-v-*-, ^=/3- 3 0^+i>-^ • (14) 



K =4t7^- 4 . T = 3tt3- 5 (15) 



The range of the attractive force is mathematically infinite, 

 but practically of the order fB~ l , and we see that T is of higher 

 order in this small quantity than K. That K is in all cases 

 of the fourth order and T of the fifth order in the range of the 

 forces is obvious from (12) without integration. 



An apparently simple example would be to suppose <f) (z) = z n . 

 From (1), (2). (4) we get 



9, — n— 4 i » 



K = j±^ (16) 



The irfrinsic pressure will thus be infinite whatever n may 

 be. If n + 4 be positive, the attraction of infinitely distant 

 parts contributes to the result : while if n + 4 be negative, the 

 parts in immediate contiguity act with infinite power. For 

 the transition case, discussed by Sutherland*, of n + 4 = 0. 



* Phil. Mag. xxiv. p. 113 (1887 



