VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. SQI 



As to the problem, of which some among them would solve particular cases, 

 in order to fix, as they pretend, its idea; it is likely they will pitch upon such 

 as are easy; for there are such in transcendental, as well as in common curves: 

 but the question is to obtain a general solution: this is no new problem; 

 M. Jo. Bernoulli proposed it in the Acta Erudit. for the month of May 1697, 

 p. 211. And as M. Fatio despised what we had done; the proposition was 

 again repeated in the Acta for May 17OO, p. 204. It may still serve to make 

 some people sensible, whether they have made such advances as we in methods, 

 and till they can arrive at a general solution, they may try their skill in fixing 

 the ideas in a particular case, which I here send you: its solution is by the 

 same M. Bernoulli : and be so good as not to give in too readily to the in- 

 sinuations of such as oppose us, when they would persuade you they found 

 no difficulty in our problem. 



A problem containing a particular case of the general problem about finding 

 a series of curves, each of which is perpendicular to another series of curves. 



Upon a right line ag (fig. 1, plate 10) as an axis, and from the point a 

 having constructed any number of curves, as abd, of such a nature, that the 

 radius osculi bo, drawn from each point of each curve, for instance from b, 

 be cut by the axis ag in the point c in a constant given ratio, so that bo may 

 be to BC, as M to n. Now the trajectorial curves enp, are to be constructed, 

 cutting the former curves abd at right angles. 



Thus far this letter, M. Leibnitz first proposed the general problem to 

 M. I'Abbe Conti in words to the following purpose : To find a line bcd, as in 

 fig. 2, that cuts at right angles, all the curves of a determinate series of the 

 same kind, ex. gr. all the hyperbolas ab, ac, ad, having the same vertex and 

 the same centre ; and that by a general method. And in the Acta Erudit. for 

 October, 1698, p. 470, 471, he calls the curves in this determinate series, 

 curvas ordinatim datas et positione datas et positione ordinatim datas. And by 

 all this the series of curves to be cut is given, and nothing more is to be 

 found, than the other series which is to cut it at right angles. But M. Leib- 

 nitz being told that his problem was solved, he changed it into a new one, 

 of finding both the series to be cut, and the other series which is to cut it. 

 And the particular problem proposed in this letter is a special case, not of the 

 general problem first proposed, as it ought to have been, but of this new 

 double problem. And the first part of this double problem (viz. by any given 

 property of a series of curves to find the curves) is a problem more difficult 

 than the former, and of which a general solution is not yet given. Mr. Leib- 

 nitz, in a letter to Mr. John Bernoulli, dated Dec. 16, 1694, and published in 

 the Acta Eruditorum for October 1698, p. 47 1, set down his solution of the 



