WAVE SURFACE TRANSFORMATIONS 133 



The following extracts from Tail's letters in March and April of 1859 

 show how thoroughly he was becoming saturated with the quaternion 

 ideas and methods. 



[March 2.] I have added a good many new theorems to the wave investigations, 

 but I fear their importance is nothing particular. 



The problem of the wave-front for which there is the greatest angular separa- 

 tion of the rays has only led me to some complicated and almost intractable 

 equations. 



I have been led in connection with the wave surface to the study of the curve 



p = $*.a, 



where p (the vector of any point) is a function of the scalar x a being a given 

 vector and </>( *) aiSi{ )bjSj( ) &c. From this I have got some curious 

 results, but have been stopped short by a difficulty of a kind new to me in 

 Quaternions, while trying to find x from 



<f> having the same meaning as before.... 



Here again a new difficulty presented itself the elimination of m (an arbitrary 

 scalar) between two equations of the form (where & = nf + <f> 3 ) 



You may see that I have my hands pretty full of work even if the matters 

 in question be of no importance. 



[March 18.] I have been working farther at the wave of late and I think am 

 in a fair way to find the equation to the central surface of the second order 

 concentric with the wave which has the closest contact with it at a given point. 

 The difficulty consists in the solution of a functional equation or rather in 

 determining the general value of a certain i/r-^o), where ^ is a linear and vector 

 function. 



I have at last attacked the subject of Potentials which was the cause of my 

 recent (and, this time, successful so far) attempt at the study of Quaternions, and 

 I think I have got the method of applying the calculus to the matter. 



I have also been working at some illustrative problems. I met with this in a 

 Cambridge Examination Paper, 'Find the locus of the centre of a sphere which 

 touches two given lines in space.' I modify it into ' Find the locus of the centre 

 of a surface of the second order, whose axes are given in ratio and direction, and 

 which touches two given lines.' 



The required locus is given in the form 



where fi and 7 are the unit vectors along the given lines, 2a is the common 

 perpendicular and <j> is the function of the surface. 



