THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE, 133 
but (v?+)I=-2nf(z,y), if (x,y) is within this area; as easily follows from the 
theorem 
Vie i | log Rf(:c’, y')dx'dy' = Inf(a, 7). 
_ The differential equation satisfied by I, together with the conditions that I and its 
first derivatives dI/dx , dI/dy are continuous throughout, define the value of the integral 
completely, and in many cases make its evaluation easy. 
(c) For example, take f(x,y) =J,,8p cos mw , with m an integer, so that 
= [ [eer J mp’ Cos mw p'dp'dw’ 
and suppose the area of integration to be a circle of radius a, with centre at the origin. 
For convenience in the ee let the imaginary part of-« be positive. 
Then [ = ae eh J,,Bpcosmo+AJ,,xpcosmw, when p<a, 
= BG,,xp cos mo, , When p>a 
A, B are determined from the conditions that I and dI/dp are continuous at p=a. 
Thus we find 
= pe —anbp cos mw + a st J nkp COS Pen oie Ka Ba —-G,KaBaS,,(Ba); (p<a) 
2 
= Fra atinne COS Mo(KAT m/ KAT mA — SmkA BAT» Ba) ; (p>a) 
By the principle of continuation in the Theory of Functions, the result is true 
whatever be the phase of x. But when the phase of « is diminished by 2z, 
G,,(kc) is increased by 277J,,(xc) 
and 
Ga(ke) by rid m (Kc) ; 
hence, equating the corresponding changes in I and its value, we obtain 
a for 
i | PAecnamen cor mar dp da’ 
0 0 
Qar 
= ae a I mkp COS MO( KAT my KAT PA = I mkABAS mB) « 
From this again it easily follows that in I and its value we may replace the G functions 
by the Y functions. 
(d) We have 
Y xp = log xpJ xp + 4x2p?-— eee 
= log «J ykp + log p(1—4K7p?...) + 4x7p?.. 
Thus logx Jocp — Yop is an integral function of «, in which 
coefficient of x? is = logp 
and coefficient of x? is 4p*logp—p?. 
