Method of Ptogressions. 325 



niula is 70 fathoms, shows that the true jDhysical state of the 

 problem has not been attained. A solution agreeing with ob- 

 servation in this element, must proceed on a law of density that 

 deviates a little from a uniform decrease of heat. The formulas 

 in the Mccaniqiie Celeste likewise fall short of observation in 

 this element: for thev give a depression of 'if)°"24 for an eleva- 

 tion of 3817 fathoms instead of 40"*25 the observed quantity, 

 b'eing at the rate of S2 fathoms to a degree. Thus, the small 

 differences between the French Tables and my formula corre- 

 spond to a difference ot no less than 12 fathoms in this element 

 of the problem. A solution that, besides fulfilling the other con- 

 ditions of the problem, should likewise agree with observation in 

 the height necessary for depressing the thermometer one degree, 

 would, in all probability, give the atmospherical refractions near 

 the horizon as much above the French Tables as these exceed 

 njy formula ; and in this manner, there is every reason to think, 

 observation would be better represented. Till such a solutioa 

 be found, it cannot be said that mathematical science has accom- 

 plished all that it is possible to do for the behoof of astronomy. 



I am, &c. 

 May 8, 1821. J. IVORY. 



LIV. Some Accoimt of a Method which may he applied to the 

 same Purposes as Sir Isaac Newton's Method of Fluxions, 

 By Mr. Thomas Tredgold. 



LETTER III.* 



On the Maxima and Minima of Quantilies. 



To the Editor. 



Sir, — In a progression of quantities, formed according to some 

 invariable law, the quantities may be of two kinds ; one of which 

 is called constant, and the other variable. 



A constant qnuntity is that which retains the same numerical 

 value in each term of a progression of quantities. 



A variable quantity is that of which the numerical value in- 

 creases or decreases in each succeeding term of a progression of 

 quantities. 



The terms of a progression may be simple, or they may be 

 compound. If the terms of a jjrfigression be compound, and 

 contain both positive and negative quantities ; and the positive 

 ([uantities be affected with the variable quantity in a manner dif- 

 ferent from the negative ones ; there will be in a progression so 



' ircc' till' iireccdintr letters at pages 177 anJ -^^^ of tliis volume. 



ctjiistitiited, 



