HO IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



Thus the square of the length of any infinitesimal arc is a homo- 

 geneous quadratic function of the differentials of the coordinates q 1 



and # 2 , the coefficients E, F, Gr 

 being functions of the co- 

 ordinates q 1} q 2 themselves. 

 If the curve is one of the 

 lines q = const , we have, 

 since dq l = J 



ds 2 2 =G-dq 2 2 , 



if it is one of the curves 

 q% = const, we have 



Considering any arc ds as the diagonal of an infinitesimal 

 parallelogram with sides ds^ and ds 2 including an angle # (Fig. 26), 

 we have by trigonometry, 



ds 2 = ds 2 -f 2 dsj_ ds 2 cos & -f- ds^. 



Making use of the above values of ds and ds 2 and comparing with 



the expression 26), we find 



F 



cos # = -= 

 1/EG 



If the coordinate lines cut each other everywhere at right angles we 

 shall have cos # = 0, F= 0, so that 



28) ds 2 = Edq l 2 + Gdq 2 2 . 



The coordinates q i9 q 2 are then said to be orthogonal curvilinear 

 coordinates. In the example above 1 ) & and (p are orthogonal, the 

 lines of constant & and cp being parallels and meridians intersecting 

 at right angles and the product term in d&dcp therefore disappearing. 

 Employing the 'expression 26) for the length of the arc, dividing 

 by dt 2 and writing 



1) We have the equations of change of coordinates, 

 x = I sin # cos qp , 

 y = I sin O 1 sin qp , 

 z = I cos -9 1 , 



from which 



- = I cos & cos qp , -- = I cos & sin op , = I sin #, 



<7aT C& Qv 



= _ ZsinO-sinqp, ~ = I sin -9 1 cos op , = 0, 



G<p Off CCp 



