WARREN'S ANTICIPATION OF HAMILTON 123 



his methods to prove almost anything. But I have given up these speculations for 

 the present till I see whether I cannot get the requisite command of Quaternions, 

 as I feel that they must inevitably much simplify the investigations.... 



All that I have to say on the subject of my School (though I fancy myself 

 rather a cosmopolitan) as regards Differentials, &c., I must beg you to let me 

 reserve for some days till I have comparative leisure again 



Yours very truly 



PETER G. TAIT. 



While Letter v was in process of construction, Hamilton sent two shorter 

 letters, Nos. vi and vn, relating to other quaternion questions. In the 

 former he discussed the surface of revolution in the form 



p = a t (j> u a.~ e , 



where < is any vector function of the scalar u, a is the vector parallel to 

 the axis of revolution and t is a second scalar variable. 



Letter vn contained an interesting historic note with reference to the 



dp 

 quantity -^ : 



" I have lately observed that Mr Warren, of Cambridge, as long ago as 1828, in his 

 Treatise on the Geom. Representation of the Square Roots of Negative Quantities,... 



gives, in his page 119, that very symbol -J- to represent a line which in length, and 



direction measures the velocity of a moving point My p has no necessary dependence 



on any sq. root of I, so long as we are merely using it to form such expressions 



as pi or -j- for the vector of velocity, ...or p" or -^ for the vector of acceleration; where 



p', p" are fairly entitled to be called ' derived functions' of t, of the 1st and 2nd 

 orders, the primitive function being p." 



In letter 7 of date October 25, 1858, Tait wrote: 



I do not intend even today to enter upon the subject of differentials though 

 I may state that I have re-read with great care your letter No. V, and have quite 

 understood, and agreed with, it while at the same time I must confess that a good 

 deal of it besides that referring more particularly to Quaternions was new to me. 



Towards the end of this letter Tait propounded the problem to find the 

 envelope of the surface S*ap + 2Sa(3p = d" when Ta= i. He had given it 

 incorrectly in a postscript to a previous letter, and Hamilton at once saw 

 there must be some mistake. Tait, after making the correction, continued 

 thus: 



The first equation represents I suppose a paraboloid and the second was 

 intended (though I presume it is not explicit enough) to mean that a might be any 



16 2 



