Mr. J. Cockle's Analysis of the Theory of Equations* 497 



tions are presented for solution, we proceed by elimination to 

 the formation of simple linear equations, each involving one 

 of the unknowns. In such equations each unknown admits 

 of one value only, and no elevation of degree is introduced by 

 elimination. If we have n linear relations among powers (or 

 roots) of n unknown quantities, then provided that the same 

 quantity is affected with the same power (or root) in each of the 

 relations, we may determine the values of the powers (or roots) 

 of each of the unknowns by linear elimination, and conse- 

 quently the unknown itself may be determined by evolution 

 or involution. These two latter terms (which might be com- 

 prehended under the common name volution) 1 should pro- 

 pose to restrict to arithmetical or, at all events, to determined 

 quantities. And I should incline to appropriate the word 

 elevation to that operation which consists in affixing a positive 

 integral index to an undetermined quantity. It is needless to 

 say what is meant by the expansion of a quantity, so elevated, 

 but by contraction we may denote the passage from the deve- 

 loped to the undeveloped form of the elevated quantity. By 

 depression we may signify the affixing a fractional index with 

 unity for its numerator and a positive integer for its denomi- 

 nator. When the indices are the same as in the previous 

 cases but negative^ we may term the respective operations ne- 

 gative elevation or depression as the case may be. When the 

 index has for its numerator and denominator integers prime 

 to one another, the operation indicated consists, of course, 

 both of elevation and depression. 



10. It is only for the purpose of calling your attention to 

 its connexion with the above considerations respecting com- 

 plex linear equations, and with the fact that, when a power is 

 known, the root may be considered as known, that I here 

 mention Varignon's solution of a quadratic. This solution 

 (see Phil. Mag. S. 2. vol. iv. pp. 314, 462, 463) is an appli- 

 cation to quadratics of the indeterminate method employed in 

 the original solution of a cubic. But the ordinary solution 

 affords ample exercise for reflection, independently of its in- 

 troducing us to the idea of multiple solutions, and the germ 

 of the theory of symmetric functions. It presents to us two 

 distinct methods ; the one capable of being rendered perfectly 

 general, the other partially so. The process common to both 

 — that of adding and subtracting the same quantity to an ex- 

 pression, and so changing its form and leaving its value un- 

 altered — may be called the method of superposition. In the 

 case of a quadratic, the quantity superposed is (I speak of the 

 ordinary process) the square of half the coefficient of the first 

 power of the unknown. This superposition is followed by a 



Phil. Mag. S. 3. No. 253. Suppl. Vol. 37. 2 K 



