about a fixed point. 45 



To my shame I must confess, that, although an occasional 

 contributor to, I am not invariably a constant reader of your 

 valuable miscellany, othennse I should not have introduced the 

 theorem in question w-ithout due acknowledgement of Professor 

 Donkin's claims to whatever merit may attach to the priority of 

 publication. The fact is, that I made out the theorem for myself 

 nine years ago, and had some communication on the subject witli 

 Professor De Morgan, v^•ho was then writing the seventeenth 

 chapter of his Differential Calculus. A recent conversation mth 

 this gentleman has brought back to my mind a \va(\. recollection 

 of the course of that com^munication. I brought under Professor 

 De Morgan^s notice the analytical memoir of Sr. Gabrio Pola on 

 the subject in the Memoirs of the Italian Society of jModeua, 

 and satisfied myself of the existence of the single axis of displace- 

 ment by compounding the two rotations in the manner given in 

 my paper, which, for the case of two arcs fixed in space, is the 

 same as Professor Donkin's, and for two arcs fixed in the rota- 

 ting body is materially, although not formally the same. 



It then occurred to me that a more simple demonstration ought 

 to be deducible from the possibility of always finding the point 

 on a sphere, by revolution about which, as a pole, one equal are 

 could actually be sho'mi to be transportable into the place of 

 another. But in proceeding to work out this idea I fell into a 

 remarkable blunder, in which I have since been followed by 

 more than one a1)le friend to whom I have proposed the ques- 

 tion. The blunder was of this kind : — Two arcs have to be 

 di'aivTi, bisecting at right angles the arcs joining the extremities 

 of two equal arcs ; the point of intersection of the two bisecting 

 arcs must in all cases fall outside the quadrilateral formed by the 

 equal and joining arcs. I supposed it to fall inside. There aj)»p 

 pears to be a fatal tendency to do so in aIL.who take the subject' 

 in hand. In consequence of this eiTor, the cause of which I did 

 not at the moment perceive, I was driven to deny and admit in 

 one breath the same proposition. Jlr. De jNIorgan sent me th6^^ 

 correct proof after this method (the same as that given by him'"' 

 at page 489 of liis Calculus), I am inclined to think after I had 

 myself detected my error; but of this I cannot feel certain. 



This is the method alluded to by me in the words "it is right 

 to bear in mind, &c.," at the time of writing which all recollec- 

 tion of the same thing having been published by Mr. De Morgan 

 had vanished from my memory. 



The proof of the triangle of rotations is so simple, that, as 

 Professor Donkin states (in a letter which he has done me the 

 favour of addressing me on the subject) was the case with him- 

 self, I thought it incredible that it should not have appeared in 

 some elementaiy work, and I was therefore at no pains to publish 



