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XITIL—On the Elimination of a, B, y, from the conditions of integrability of S uadp, 
SuBdp, Suydp. By G. Piarr, Docteur és-sciences. Communicated by 
Prof. Tart. 
(Read December 21, 1874.) 
The proposed scalars represent the differentials of the three functions, 
X, Y, Z, of the co-ordinates z, y, z, put under the form 
dX =—S<1Xdp, dY = etc, 
where <j represents the operator 
ae a ke ad 
“de tI dy tae? 
and where 
dp = idx + jdy + kdz. 
These functions, equated respectively to constants, are intended to represent 
a system of orthogonal surfaces. The conditions to which X, Y, Z must for this 
end satisfy, are represented by 
X= Ya, aN — 78), IZ = Wy, 
where w designates the tensor which the three expressions must have in com- 
mon, and where a, 8, y designate a system of treble rectangular unit-vectors ; 
and w, as well as a,8,y, being generally variable from one point of space to 
another, a, 8, y satisfying at every point to the nine relations : 
= 8) Sy aa Sa Bi= oy, ete. ete. 
The possibility of the expressions wa, etc., for IX, etc., is expressed by 
the conditions of integrability of the proposed scalars, and as these conditions 
consist in three relations between w, a, 8, y, and as the vector relations between 
a, B, y, give the required “ fourth” equation, there is no impossibility @ priori 
of conceiving the existence of an equation containing w alone after the elimina- 
tion of a, B, y. 
‘In Professor Tarr’s paper “On Orthogonal and Isothermal Surfaces” 
(Trans. R.S.E. 1873-74), the problem of elimination is proposed and solved, but 
at the opening of the paper we read of the suggestion, that quaternion methods 
require improvements in their application to the proposed problem of elimination. 
In the present paper we give an account of the attempts we have made to 
follow out the suggestion quoted. Our plan has been to define certain quater- 
nion functions of a, 8, y, to which the question leads to, but which may present 
VOL. XXVII. PART III. 3 U 
