■174 Proceedings of the Royal Irish Academy. 



Substituting we get from (8) 



A = - SSjOip.f - (f + JFft)((ri + (7o) + I VMVM{a^ + (y^)T^i<j, - a^) 



+ (<Ti - (To) >S'((7i - a^ (f + i Fft) ((7i - (72) + -I Fw Fw ((Ti - (T2))r'='((7: - (7c) 

 + (o-i - (7o)aS'Fw((Ti - (72) (r + iFw((7i + (72))r-2((7i - (73) 

 + F((7, - (T2) F(t Fw ((71 - (72) + F(0(T2 Fw(Tj) r-^((7i - (72). 



In like manner we find 



IX = - SSfiieal - i F(((72 + (7i)t + (72 F&)(Ti + (7i Fa)(73 + (l) Fw ((7i + (72)) T'\<Jx - (72. 



+ F<7i(72>S'(r + i F(y ((Ti + (7.) + i Fa> F(o (^n + (72)) ((Ti - (72) r-n<yi - <^2) 



- F(Ti(72^w((7i - (72) (f + |Fw((7i + (T3)) :?^-^((7i - a2) + (F((7i - (72)f 

 + Fa)F(7i(72) (>St((72 - (7i) + /S'tOCTsCTi) 7^'^((7i - (73). 



In both the above expressions the summation extends to each pair of 

 particles once. 



If we introduce three linear vector functions which we may call respectively 

 the^rs^, second, and third inertia functions defined thus 



(^,(a) = S26ie2(«r-X(7i - (T2) - ((71 - C72)/S'((7i - a^)aT-\<y, - (72)) 



(/)2(a) = S26i62(|Fa((7i + (72)r-l(c7i - a^ - ((7i - (72)/S'(72(T,ar-^((7i - (72)), 



and its conjugate 



- <^^{u) = 2261^2(1 F(,7i + (72)aT-l(<7i - (T2) - F(72(7i>S'((7i " f7.:)aT-\a- (72)) 

 ^3(0) = S2eie2(|((T2Ftt(7i -h (7iFa(73):7'"^((7i - (72) - F(7i(72*Sa(7l(72r"^ ((7i - (Tj), 



we can express A and /z as follows : — 



A = ^i(f) + Fw^i( ') + ^i(FTa>) + ^2(«) + F(i>^2((o), 

 ju = FrA + FT(^i(r) + (^2(w)) + (^2X1^) t- <^3(w)- 



In the notation of the last paragraph the equations of motion may be 



written 



^m^ + A = A'' 



If we regard the mass of each particle as zero, these equations are simply 



A = A'' . 



jU = JU . 



We proceed to consider these, afterwards considering the effect of the masses 

 of the particles. 



If ^{r\ =-. SaiS'/Sr where the various vectors «, /3, etc., are of constant 



