1867.] On the Orders and Genera of Ternary Quadratic Forms. 387 



than if they retained their transversal positions. Two levels are fixed to 

 the horizontal-axis frame. The instrument is duly counterpoised. 



The general arrangement of the instrument may be best conceived by 

 supposing an equator eal of the German form to be adjusted as it would be 

 at the pole, when its polar axis would represent the vertical axis of the 

 zenith sector, and its declination axis the horizontal axis. 



The instrument has been too short a time in my hands to admit of ray 

 forming an opinion as to its probable success. I anticipate some advantage 

 from the arrangement of the sectors, which, being in identical circum- 

 stances, should be liable to no inequalities of either flexure or temperature. 



Both of the principal axes of the instruments are provided with inde- 

 pendent adjustments, the action of which appears to be very satisfactory. 



In concluding this brief and imperfect notice, I beg to state that I hope 

 to draw up hereafter a full and detailed description of all the instruments 

 given in the foregoing list. At present my time is absorbed by the trials 

 and experiments to which it is necessary that I should subject them before 

 their despatch to India ; and I trust the Royal Society will accept this 

 explanation as an excuse for the meagreness of the present account. 



XL " On the Orders and Genera of Ternary Quadratic Forms.^^ 

 By Henry J. Stephen Smith, M.A., F.R.S., Savilian Professor 

 of Geometry in the University of Oxford. Received February 21, 

 1867. 



(Abstract.) 



The object of this Paper is to supply demonstrations of the undemon- 

 strated results, relating to Ternary Quadratic Forms, which are contained 

 in an im.portant memoir of Eisenstein's ("Neue Theoreme der hoheren 

 Arithmetik," Crelle's Journal, vol. xxxv. p. 117), — and, at the same time, 

 to extend those results to the cases not considered by him in that Memoir. 

 The following are the principal points in which the theory of Eisenstein 

 has been thus further developed : — 



1. In Eisenstein's Memoir forms of an even discriminant only are con- 

 sidered. Such forms, and their contravariants, are always properly primi- 

 tive ; they have particular generic characters with respect to uneven primes 

 dividing the discriminant, but have no supplementary characters {i. e. cha- 

 racters with respect to 4 or 8). The case of forms of an even discriminant 

 is more complicated. Besides the properly primitive order, there may 

 exist, in this case, an improperly primitive order, in which the forms 

 themselves are improperly primitive, and their contravariants properly 

 primitive, — or, again, an improperly primitive order, in which the forms 

 themselves are properly primitive, and their contravariants improperly pri- 

 mitive. Further, forms of an even discriminant may have characters with 

 respect to 4 or 8 ; and a complete enumeration of these supplementary 

 characters requires a careful distinction of cases. To facilitate this enu- 



VOL. XV. 2 I 



