68 PHILOSOPHICAL TRANSACTIONS. [aNNO 1786. 



greatest from the given or assumed hypothesis = O, and from the resulting equa- 

 tions may be found the first approximates ax", A'a?", &c. either accurately or 

 nearly ; then, in the given equations for y, 2, &c. write ?/' -f (a -|- a) x" -{- 

 ^x"-^"' -f- &c. z 4- (a' + a) x"" 4- Bx'"+"'', where a, a', &c. are very small quan- 

 tities; and suppose the terms of each of the equations which become greatest 

 from the above-mentioned hypothesis respectively = 0, and from the equations 

 resulting deduce the quantities a, a, &c. ; w', rn, &c. ; b, b', &c. ; and so on : 

 or assume 2/ = (a + la + al + &c.) x" -\- (b -\- lb + bl -{■ &cc.)x''+"' -f &c. ; 

 z = (a' -f la -\- a I -f- &c.) X'" -|- (b' -f 1^' + b'l + &c.) x'"+''' + &c. &c. ; 

 substitute these quantities for their values in the given equations, and from 

 equating the correspondent terms of the resulting equations may be deduced the 

 quantities required. 



The differences of the indexes n, &c. m\ &c. may be deduced by writing 

 X", x^y &c, for 7/f 2, &c. in the given equations, from the differences of the in- 

 dexes of the quantities resulting. The same principles may be applied in finding 

 the above-mentioned differences, when two or more values are ax", &c. which 

 were applied in a like case to one equation having two unknown quantitieSi 

 The same principles which discover the cases in which a series deduced from an 

 equation having two unknown quantities will converge, may be applied for the 

 same purpose to these series. 



In finding the series which expresses the value of y in terms of x, there will 

 always occur as many invariable quantities to be assumed at will, as is the order 

 of the fluxional equation, provided the series begins from its first terms ; and to 

 find them there will result equations easily reducible to homogeneous fluxional 

 equations, of which the orders do not exceed m. 



V^. Experiments on Hepatic Air.* By Rich. Kirwany Esq. F.R.S. p. 118. 

 Hepatic air is that species of permanently elastic fluid which is obtained from 

 combinations of sulphur with various substances, as alkalis, earths, metals, &c. 

 It possesses many peculiar and distinct properties ; among which the most ob- 

 vious are, a disagreeable characteristic smell emitted by no other known sub- 

 stance ; inflammability, when mixed with a certain proportion of respirable or 

 nitrous air; miscibility with water, to a certain degree; and a power of dis- 

 colouring metals, particularly silver and mercury. These properties were first 

 discovered by that incomparable analyst Mr. Scheele. This air acts an important 

 part in the economy of nature. It is frequently found in coal-pits ; and the 

 truly excellent and ever to be regretted M. Bergman has shown it to be the prin- 

 ciple on which the sulphureous properties of many mineral waters depend, and 

 thus happily terminated the numerous disputes which the obscurity of that 



* Termed sulph'aretted hydrogen gas, in the new chemical nomenclature. 



