1074 Proceedings of Royal Society of Edinburgh. [sess. 
§ 81. For the three-dimensional system let, in fig. 31, if/ he 
the inclination to 0 X of the rearward wave-normal of one of the 
constituent systems of waves. This is also the inclination to 0 Y 
of the medial line of the travelling forcive to which that set of 
waves is due. Take now for the forcive obtained by the super- 
position of an infinite number of constituents, as described 
in § 80, 
-1%, y)= f’ctyfr — ^ ^ . . (108), 
g y y J o r [(x cos xf/ + y sm xj /) 2 + b 2 ] 
where k may be a function of xf/, and b is the same for all values 
of \f/. 
For the case of a circular forcive system we must take k 
constant ; and we find 
- II(r) = 7r ^ . — wher er 2 = x 2 + y 2 . . (109). 
9 Jif + b 2 ) 
§ 82. Let now the forcive, whether circular or not, be kept 
travelling in the direction of x negative * with velocity v : and 
* This is opposite to the direction of the motion of the forcive in fig. 26 . 
