186 Prof. D. N. Mallik on Lines of 



Consider, first, the line 



r + s + t = 2q % . 

 The equation is 



q x cos X + q 2 cos 2 + q 3 cos 3 = q x + q 2 + ? 3 — 2? 3 

 or ?i (cos X — 1) + q 2 (cos 2 — 1) + ? 3 (cos 3 + 1) = 0. 

 Putting 0, = 0, 2 = *t? we nave 



- 2? 2 -H q z (cos 3 + l) =0, 



or cos #3 = -^ — 1, 



giving the direction of the tangent at q 3 or! the line. 



[I£ q 2 = q dJ 3 = O, i.e., the line coincides with the axis.] 



Again putting 6 2 = 0, 3 = ir, 

 ?! (cos 0,-1) = 0. 

 0! = ; 

 also 6 X = 0, 3 = 7r, gives 2 = 0. 



Thus we see that the axis joining e x and e 2 is a portion of 

 the line of forco defined by r + s + t = 2q 3 . And we conclude 

 that this limiting line of force consists of a curved line 

 through q s and the axis joining q l9 q%. 



Taking next the line 



r + s + t = 2q z +2q 2 , 

 we have 



q x (cos 0i — 1) 4- q 2 (cos 2 — 1) + q s (cos 2 + 1) = 0. 



Putting 2 = 0, 3 = 7T, 



we have q x (cos l — 1) = — 2q 2i 



or cos 0, = 1 — , 



giving the direction of the tangent to the line at q x . 

 Again putting X = 0, 2 = w, 

 we have 3 = 7r 



and if 6 X = 0, 3 = tt, we have 2 = 0, 



showing that the axis joining e 2 , e z is a portion of the line of 

 force, defined by r + s + t — 2q 3 + 2q 2 . 



Hence this limiting line of force consists of a curved line 

 through q± and the axis joining ? 2 , q B . 



