11^ Prof. Challis's Proof that every Equation 



hammer in the hand, the convexity of the strata is manifested 

 sometimes in one sense, sometimes in the contrary, according 

 as one or the other of these two cm-rents predominates. 



It is well to observe that the conductors are electrified by in- 

 fluence in these experiments. This is proved by insulating 

 them, and making them communicate with an electroscope. 

 Thus the tinfoil, when one of the poles is active and the second 

 extremity of the induced wire is insulated, gives to the electro- 

 scope an electricity like that of the active pole. If the insulated 

 extremity of the induced wire touches the tinfoil, the electroscope 

 is charged with the electricity given by this wire to the con- 

 ductor : the experiment must be made with attention, for the 

 electricity appears alternately to approach and recede from the 

 electroscope. This instrument may also be charged with the 

 tinfoil when the two poles of the tube are active. 



XVI. A Proof that every Equation has as many Roots as it has 

 Dimensions. By Professor Challis*. 



IT will probably be conceded that this theorem, which has 

 been the subject of so much discussion, will receive a legi- 

 timate proof, if a method, however operose, be indicated, by 

 which as many roots of any numerical equation as there are di- 

 mensions can be actually found, whether they be possible or 

 impossible, and at the same time it be shown that no other 

 quantities are roots. It might perhaps be questioned whether 

 any other kind of proof does not involve a petitio princijiii. 



The method I am about to propose rests on the following 

 algebraic principles. Algebra consists of two parts, the distinc- 

 tion between which is seldom marked with sufficient clearness in 

 algebraic treatises. One part is wholly concerned with rules of 

 operation and the generation of different kinds of symbolic 

 representation of quantity, and is preparatory to, and indepen- 

 dent of, the other part, which entirely consists in the formation 

 and solution of equations. In the former the sign of equality 

 means identity of the functions it separates, under difference of 

 forms ; in the latter the same sign means equality of value for 

 certain values of an unhioivn quantity. These two significations 

 might with advantage be distinguished by a difi"erence in the 

 sign. One of the most important and general results deducible 

 from the operations of the first part of algebra is, that every 

 function can be reduced to the form A + B-v/ — 1, A and B 

 standing for algebraic functions which en dcrniere analyse are 

 positive or negative numerical quantities. Now when in form- 

 * Commimicated by the Author. 



