438 Mr. J. Cockle on Algebraic Transformation, 



where s, t, u, v, which are all real, satisfy the relation 



{s±u±vf-\-f=\, 

 the signs being regulated by the form of modulus selected. And 

 we are at liberty to assume that 



s + u + v= cos jo, and t = sin p. 

 (The angle j» is the amplitude of the coquaternion.) And if we make 



s = qcosp, u = rcosp, and v= ±(1 — q + r)cosp, 

 any coquaternion may be put under the form 



M{qCOsp + cismp + (3rcosp + y(l — q + r) cosjo}. 



It is to be observed that (1) the modulus of the product of two 

 coquaternions is the product of the moduli of tbe factors, and 

 (2) the amplitude of the product is the sum of the amplitudes 

 of the factors. I have already given the corresponding expres- 

 sion of a tessarine (S. 3. vol. xxxvi. p. 290). Each system of 

 multiple algebra may — like tbe quaternion theory with reference 

 to geometry — be the appropriate exponent of some branch of 

 science, or find some peculiar application in geometry itself. 



I am indebted to Professor W. F. Donkin for pointing out to 

 me, in a letter dated Keswick, August 29, 1850, that " it is easy 

 to give an interpretation to the [tessarine] symbols considered 

 as representing transposition ; thus, according to Sir W. Hamil- 

 ton's notation, we should have 



i== ^-l,0, -3, 8> i = -^2,3, 0, 1' ^ = ^-3, 2,-1,0- ' 



Professor Donkin adds, that " perhaps this might suggest a geo- 

 metrical interpretation." Should such an interpretation be sa- 

 tisfactorily arrived at and acceded to, the normal nature of the 

 tessarine system would afford a great practical convenience. 



Let 0, tbe origin of coordinates, be the centre of a sphere 

 whose pole is A and radius unity, and which is intersected in B 

 by the radius vector of a tessarine. Through B draw a great 

 circle meeting the first meridian in C, and such that the angle 

 ABC is equal to tt — 6; then 



cos C = cos 6 cos -ty+ sin 6 sin -»|r cos <f>, 



and the equation for the submodulus (S. 3. vol. xxxiv. p. 47) 

 may be expressed as follows, — 



v 2 =^, 2 sin sin $ cos C. 



The new modulus and amplitude (Ibid. vol. xxxvi. p. 291) will 

 however probably supersede the former expressions. 



(c.) If, in certain of my previous researches on equations (Ibid, 

 vol. xxvii. p. 293), we denote by p the radical in (18) the re- 

 maining coefficients of (16), viz. y 4 , y b , . . and 7 10 (misprinted 



