5^ PHlLOSOPHICAt TRANSACTIONS. [aNNO I693. 



theorem -, — ^-^— — the term p r a vanishes, as being finite, which is 



dr+dp — prf ^ ^ ° 



no part of the other infinite terms, and dividing the remainder by the infinite 



part d, the theorem will stand thus -^^ =f, or in glass, -7-^- =f.. 



In case the lens were plano-convex exposed to diverging beams, instead of 



_ L.-LL r being infinite, it will be -^—— = f, or :; — ^ =/'ifthe 



dr+df — prf' ° ' ^ — Pt d - 2 f -^ 



lens be glass. 



If the lens be double-convex, and r be equal to p, as being formed of seg- 

 ments of equal spheres, then will . , . be reduced to — ^ — — = /; 



T r ' dr + df — prf i d — p r "^ 



and in case d be infinite, then it will yet be farther contracted to i /» r, and p 



being = , the focal distance in glass will be = r, in water 14^ r, but in 



diamond 4- r. 



I am sensible that these examples are too much for the complete analyst, 

 though I fear too little for the less skilful, it being very hard, if possible, in 

 such matters, so to write as to give satisfaction to both ; or to please the one, 

 and instruct the other. But this may suffice to show the extent of our theorem, 

 and how easy a reduction adapts any one case to all the rest. Nor is this only 

 useful to discover the focus from the other proposed data, but from the 

 focus given, we may thereby determine the distance of the object, or from 

 the focus and distance given, we may find of what sphere it is requisite to take 

 another segment, to make any given segment of another sphere cast the beams 

 from the distance d to the focus /; as likewise from the lens, focus, and dis- 

 tance given, to find the ratio of refraction, or of m to w, requisite to answer 

 those data. All which it is obvious, are fully determined from the equation we 

 have hitherto used, viz. pdfr = drf+dff—prf/; for to find d, the 



theorem is , . ^ V = d, the distance of the object. 



rf+ff-Pfr ' •' 



For «, the rule is — j — T^r-, . = ». 



^' pdr + df+ prf '^ 



But for p, it is -^^ — -^ = p, which latter determines the ratio of refrac- 

 ^ dfr+ffT ^' 



tion, m being to n as 1 -|- jb to p. 



I shall not expatiate on these particulars, but leave them for the exercise of 



those who are desirous to be informed in optical matters, which I am bold to 



say are comprehended in these three rules, as fully as the most inquisitive can 



desire them, and in all possible cases ; regard being had to the signs -\- and — , 



as in the former cases of finding the focua. I shall only show two considerable 



