604 PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



irrationality will be contained in the correspondent values of each of the un- 

 known quantities, unless two or more values of one of them are equal, &c. 

 The same observation is also applied to the co-efficients of an equation deduced 

 from a given equation. — 20. In the Miscell. was published a new method of ex- 

 terminating, from a given equation, irrational quantities, by finding the multi- 

 pliers which multiplied into it give a rational product. — 21. In the Medit. 1770, 

 are given the different resolutions of a certain quantity (a 2 + rb'-) im +' and 

 (a 2 + rZ> 2 ) 2, "+ 2 into quantities of the same kind. — 22. Mr. De La Grange has 

 very elegantly demonstrated Mr. Wilson's celebrated property of prime numbers 

 contained in my book. In the last edition of the Medit. the same property is 

 demonstrated, and some similar ones added. 



23. In the Miscell. is given a method of finding all the integral correspon- 

 dent values of the unknown quantities of a given simple equation, having two 

 or more unknown quantities ; and, in the Medit. 1770, are given methods of 

 reducing simple and other algebraical equations into one, so that some unknown 

 quantities may be exterminated ; and if the unknown quantities of the resulting 

 equations be integral or rational, the unknown quantities exterminated may 

 also be integral or rational. — 24. In the Medit. are given rules for finding the 

 different and correspondent roots of an equation, whose resolution is given. — 25. 

 Mr. De La Grange has recommended my new transformation of equations, pub- 

 lished in the Miscell. which perhaps is not less general nor elegant than any yet 

 published; and in the Meditat. 1770 is given a method very useful in finding 

 the co-efficients. 



If either here, or in the preface to the Medit. Algebraicae, I have ascribed to 

 myself any algebraical, or in the properties of curve lines any geometrical, or in 

 the Medit. Analyt. any analytical invention, which has been before published by 

 any other person, I can only plead ignorance of it, and shall on the very first 

 conviction acknowledge it. I must further add, that I have been able to carry 

 my algebraical improvements into geometry ; for from them, with some geo- 

 metrical principles added, I have, unless I am deceived, deduced as many new 

 properties of conic sections and curve lines as have been published by any one 

 since the great geometrician Apollonius. 



XXIX. Of a Remarkable Frost on the 13d of June, 1783. By the Rev. Sir 

 John Cidlum, Bart., F. R. S., and S.J. p. 41 6. 

 About 6 o'clock, that morning, says Sir J. C., I observed the air very much 

 condensed in my chamber-window; and on getting up was informed by a tenant, 

 who lives near, that finding himself cold in bed, about 3 o'clock in the morning, 

 he looked out at his window, and to his great surprize saw the ground covered 

 with a white frost; and I was afterwards assured, on indubitable authority, that 



