FHOCEEDINGS OF THE AMERICAN ACADEMY 



where 



cr^ = A^ + fji' -\- 2 \ fjL COS K a, (35) 



. _i ( u sin K a ) /oo\ 



T = sm N — ^ >■ . {6b) 



'- V -^^ + M^ + 2 \fi cos Ka^ 



This is a motor of the same kind as L and 31. Its strength is a, and 

 its axis makes with the axis of x the angle - . In the case of doublets, 



K 



that is, when k = I, this result may be put into a simple form. If 

 two doublets L and M exist together at a point 0, and if the direc- 

 tions of the two straight lines OA, OB, show the directions of the 

 axes of L and M respectively, and the lengths of OA and OB the 

 strengths of L and 31 on some convenient scale, then the direction of 

 the axis of the result of L and M will be given by the directions and 

 the strengths of the resultant bv the leno-th of the diagonal of the 

 parallelogram of which OA and OB are adjacent sides. Doublets 

 then can be compounded and resolved by compounding and resolving 

 their axes like forces or velocities. 



17. Let there be any motor J/ at a distance a from the centre of 

 a circumference of radius r drawn in a plane, and let the radius 

 drawn through 31 be taken as axis of x. Let n . x {x, y-, a) be the 

 flow function due to M, and its image in the circumference, existing 

 together. If another motor B, of the same kind and strength as A 

 but of opposite sign, exist at a point on the axis of x at a distance 

 a + Aa from the centre, the flow function due to B and its image 

 together is 



- /* • X (^» y' « + -^ «)• 



The flow function due to A, B, and their two images existing 

 together is, therefore, — /^t . A„ ^ (x, y, a). 



If now Aa be made to approach zero and ^ to increase so that 

 H . Aa is equal to the constant X, we get as the flow functions due to 

 the resulting new motor at A and its image in the circumference 



.// = - X . z>„x (^, y^ «)• (3") 



F (2, n), of which i// is the real factor of the imaginary part, is con- 

 nected with/(2, a), of which ^ is the real factor of the imaginary part 

 by the equation 



F{z,a)=-\ D„f{z,a). (38) 



