136 PETER GUTHRIE TAIT 



THEOREM VI. If these roots be also all unequal, then the eq M , 

 (4>+gi)Pi = 0, (<+#,) Ps = 0, (<t>+g,)p s = 0, (97), 



are satisfied by the 3 rectangular directions p lt p t , p, of Theorem iv, and by those 

 directions (or their opposites) only. 



THEOREM vn. For any other vector, p=Xipi+Xip,+x0t, (6), 

 we have <f>p = - (g^x lpl +g*x*p t +g,x,p s \ (t), 



and Sp<f>p = - (g&p? +g*x?p? +g t xfpt) > (K). 



THEOREM VIII. Whatever the real scalar, g, and the real vectors, a, a',... and 

 /3, /S', ... may be, it is possible to find 3 real scalars, gi,g^;g 3 , and 3 real and rectangular 

 unit vectors, p lt p t , p,, such that the following shall be an identical transformation : 



THEOREM IX. The data, g, a, , a, ft, ... being still real we have finally this 

 other transformation : 



gf + USapSpp =? + 2S<tpSpp, (ft), 



without any sign of summation in the 2nd number ; and g~, a, $\ can always be 

 made real. 



Having written so far, and even had the first sheet of this letter copied (into A. 

 1859), I think that I may now indulge myself in opening your letter received this 

 morning.... For I have been apprehensive of your anticipating me, or hitting on my 

 old train of thought, before I had (as above) recovered it for myself. 



Tait, on April 21, replied: 



I was greatly pleased with the transformations in XIX. I can easily prove all 

 your theorems with the exception of the first, i.e. that " Vp<f>p = o admits of one real 

 solution at least." It is certainly a very elegant mode of attacking the question, and 

 I had never thought of so simple a point of view as the making the normal coincide 

 with the radius vector. But when I try to prove your theorem, I fall back again 

 into the cubic of my letter 1 30, or at all events a simple case of it, so that I do 

 not see how you manage to avoid a reference to something or other equivalent to 

 i,j, k. 



In a PS. to letter XXH, dated Easter Tuesday, 1859, Hamilton 

 indicated the proof which Tait longed for : 



" My Theorem I, of Letter xix, was proved by showing, on the plan of Lecture vn, 

 Art. 567, that the equation 



could be satisfied without our having also p = o, provided that g was a root of a 

 certain cubic equation. It is not at all necessary, for this purpose, that < should 

 satisfy the functional condition 



1 In regard to letter 30 Hamilton had remarked that he liked the look of it. Unfortu- 

 nately a copy of this particular letter does not seem to have been preserved by Tait. 



