110 Prof. H. J. S. Smith — Criterion of Resolubility, ^c. [Jan. 28, 



minutes past eight it increased from 16 to 43 lbs. in the short space of two 

 or three minutes ; the barometer, being at its minimum, suddenly rose about 

 three-hundredths of an inch, and during the heaviest part of the storm it con- 

 tinued to rise at the rate of about one-tenth of an inch an hour. The oscil- 

 lations in the mercurial column, as will be seen by the diagram, were large 

 and. frequent during tlie storm, one of the most remarkable being imme- 

 diately after 10^ a.m. and nearly coincident with two of the heaviest gusts 

 of wind ; the depression in this case amounted to between four and five 

 hundredths of an inch, tlie rise following the fall so quickly that the clock 

 moved the recording-cylinder only through just sufficient space to cause a 

 double line to be traced by the pencil. 



III. On the Criterion of Eesolubility in Integral Numbers of the 

 Indeterminate Equation 



/= + a'x'^ + aV^ + 2ba:'a;" + -f 35% = 0.^' 



By H. J. Stephen Smith, M.A., F.R.S., Savilian Professor of 

 Geometry in the University of Oxford. Received January 20, 1864. 



It is sufficient to consider the case in which / is an indefinite form of a 

 determinant different from zero. "We may also suppose that / is primitive, 

 i. e. that the six numbers a, a', a", b, b\ b" do not admit of any common 

 divisor. We represent by O the greatest common divisor of the minors of 

 the matrix of /, by AO^ the determinant of /, and by i2F the contravariant 

 of/, ^. e. the form 



(62-«V/>'^+ ; 



ilA^ will then be the determinant of F, and A/ its contravariant. By 

 il. A, and ^2A we denote the quotients obtained by dividing ^1, A, and OA 

 by the greatest squares contained in them respectively ; w is any uneven 

 prime dividing but not A ; 3 is any uneven prime dividing A, but not 12; 

 and Q is any uneven prime dividing both 12 and A, and consequently not 

 dividing 12 A. We may then enunciate the theorem — 



" The equation /= will or will not be resoluble in integral numbers dif- 

 ferent from zero according as the equations included in the formulae 



(?)-©■ (?)-©■ 



are or are not satisfied." 



The symbols ^y^, and are the quadratic symbols of 



Legendre; the symbols ^y^, ^^j, are generic characters of / 



(see the Memoir of Eisenstein, " Neue Theoreme der hoheren Arithmetik," 

 in his ' Mathematische Abhandlungen,* p. 185, or in Crelle's Journal, 

 vol. XXXV. p. 125), 



The theorem includes those of Legendre and Gauss on the resolubility 



