102 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



inclined at an angle n times as great. Compare Art. 80. If we 



*=(r ~ (54), 



then make n = - , so that 



OL 



a lune of angle a having its angular points at c lt C 2 becomes in the 

 plane of Z one straight line, and therefore by (53) in the plane 

 of Z a circle. The constants in (53) may be so determined that 

 any three points on the perimeter of the lune correspond to any 

 three points in the plane of Z . 



Since, in the same way, the assumption 



^=fe$ 5 (55) 



transforms a lune of angle a having its extremities at c/, c/ into a 

 straight line in the plane of Z , we see that (53), with (54) and 

 (55), transforms a lune of angle a having its extremities at c lt c 2 

 into a lune of angle a having its extremities at c/, c 2 , provided the 

 ratios A : B : C : D be (as they may be) so chosen that three arbi 

 trary points on the perimeter of the one correspond to three arbi 

 trary points on the perimeter of the other. This is the solution 

 of the problem above stated. 



In the hydrodynamical application one of the lunes is the strip 

 of the plane of w bounded by the straight lines i|r = 0, ty = b. In 

 this case the expression corresponding to the right-hand side of 

 (54), (with w written for z) assumes an indeterminate form; the 

 angular points of the lune being at infinity, whilst its angle is 

 zero. When evaluated by the usual methods this expression be- 



1TW 



comes e 6 . It is in fact obvious from Art. 78 that the assumption 



Y=e* ........................ (50) 



converts the two straight lines in question into the one straight 

 line Y= 0, and therefore serves the purpose of (53). 



We proceed to give the more important of the applications of 

 the above method which have been made by Kirchhoff. 



