54S PHILOSOPHICAL TRANSACTIONS. [ANNO 1 7QQ. 



follows that no biquadratic having all its roots real, can admit of a real solution by 

 either of these methods. 



32. The formula expressing the actual resolution of a biquadratic has not been 

 given; the writers on algebra going no further than to point out the cubics by 

 means of which such a resolution may be obtained. To be sure, such a formula 

 would be very long, and, till the imperfection in the cubic resolution, which must 

 make a large part of it, can be removed, embarrassed with radicals, so as to be of 

 little practical use; but it would be a valuable accession to' the theoretical part of 

 algebra, to have the analysis of this degree carried as far as that of the preceding, 

 by developing every part of the functions that enter into the resolution, so as to 

 be able to compose it at once, or make a complete reduction of the equation, 

 without the intervention of any other steps. 



33. Let n be taken = 5, or any higher number. Here the number of dif- 

 ferences is increased to 20; and the higher we go the more they increase, so that 

 a direct simple resolution is out of the question. Nor are we yet acquainted with 

 any peculiarity attending 5, or any higher number of quantities, on which we can 

 ground a relation to effect a reduction of any sort; therefore no method is known 

 for equations of this and the higher orders. Whether any may ever be discovered, 

 it is not easy to pronounce: if the reasoning from art. 8 to art. 15, of this paper, 

 be correct, there can be no chance, until some peculiar property of quantities of 

 this class can be hit on. It is perhaps a discouraging presumption against the ex- 

 istence of any such property, that no art or labour has hitherto afforded the least 

 clue to lead to one; but, on the other hand, it is impossible to tell what general 

 properties of quantity may remain to be discovered; and, from the great distance 

 the peculiarities of the degrees we have treated of lie from the surface, and their 

 total want of order or connection with each other, it may be justly expected that 

 those of the higher degrees may lie still more detached and remote, beyond any 

 efforts that have yet been made on the subject. The proper method to proceed 

 seems therefore to be, abandoning all projects for the general resolution of equa- 

 tions, to investigate regularly the abstract properties of each separate order or 

 number of quantities, turning them into all shapes, sifting all their combinations, 

 and constructing and examining the equations of different complex functions of 

 them, in order to see if latent peculiarities be not to be traced out in some of 

 them. Wherever any distinguishing property is found, it will, by the principles 

 here explained, infallibly lead to some method for the degree to which it belongs; 

 and whoever may be fortunate enough to discover any such property, in 5, 6, or 

 any higher order of quantities, will have the honour of removing the important 

 and hitherto impenetrable barrier, which has so long obstructed the further im- 

 provement of algebra. 



XVIL On different Sorts of Lime used in Agriculture. By Smithson Tennant, 



Esq., F. R. S. p. 305. 

 It is said, that in the neighbourhood of Doncaster, there are 2 kinds of lime 



