852 Mr. J. Bose-Innes on the Motion of a Pendulum 



be offered of this integral, which, it is believed, is rather 

 simpler than any hitherto proposed. 



We may recall the well-known connexion between the 

 independent variable x of the above integral and the radical 

 occurring in the denominator of the subject of integration, 

 viz. : — that ^1 — tc 2 sin 2 x represents the cosine o£ a side of 

 a spherical triangle opposite to an angle of value x, provided 

 that the constant ratio of the sine of a side to the sine of the 

 opposite angle is k. We may conveniently introduce a 

 simplification by assuming that one of the angles of the 

 triangle is a right angle. 



Consider then a spherical triangle ABC of which the angle 

 at C is a right angle ; and let sin AB be denoted by k. 



Then 



sin BC 



: A~ — K ' 



sm A 

 and cosBC = \^1 — a: 2 sin 2 A. 



The fundamental elliptic integral may be therefore written 



"* dk . 



c 



cos CB 



But since C is a right angle we have 



cos AC x cos BC = cos AB 



so that the integral may be also written 

 p cos AC d A 



Let AC' be a neighbouring position of the arm AC, and 

 produce AC and AC 7 to meet the circle whose pole is A in 

 the points D and D' respectively. Let the radius of the 

 sphere on which the triangles are drawn be denoted by R. 



