Prof. Cayley on the Mathematical Theory of Isomers. 445 



a point 0, C, &c. as the case may be, but not to another 

 point H), we may without breaking up the diagram remove all 

 the points H with the links belonging to them, and thus obtain 

 a diagram without any points H : such a diagram may be termed 

 a " kenogram :" the valency is obviously that of the original 

 diagram plus the number of removed H's. 



If from a kenogram we remove every point 0, C, &c. connected 

 with the rest of the diagram by a single link only (each with the 

 link belonging to it), and so on indefinitely as long as the pro- 

 cess is practicable, we arrive at last at a diagram in which every 

 point 0, C, &c. is connected with the rest of the diagram by two 

 links at least : this may be called a " mere kenogram. - " 



Each or any point of a mere kenogram may be made the 

 origin of a " ramification ;" viz. we have here links branching out 

 from the original point, and then again from the derived points, 

 and so on any number of times, and never again uniting. We 

 can thus from the mere kenogram obtain (in an infinite variety of 

 ways) a diagram. The diagram completely determines the mere 

 kenogram ; and consequently two diagrams cannot be identical 

 unless they have the same mere kenogram. Observe that the 

 mere kenogram may evanesce altogether; viz. this will be the 

 case if the diagram or kenogram is a simple ramification. 



A ramification of n points C is (2/i + 2)valent : in fact this is 

 so in the most simple case n= 1 ; and admitting it to be true for 

 any value of n, it is at once seen to be true for the next suc- 

 ceeding value. But no kenogram of points C is so much as 

 (2» + 2)-valent; for instance, 3 points C linked into a triangle, 

 instead of being 8-valent are only 6-valent. We have therefore 

 plerograms of n points C and 2n + 2 points H, say plerograms 

 C H 2n+2 ; and in any such plerogram the kenogram is of ne- 

 cessity a ramification of n points C ; viz. the different cases of 

 such ramifications are* 



?i=l. ra=2. n = 3. » = 4. 





v 



W («) (■) W 08) 



* The distinction in the diagrams of asterisks and dots is to be m the 

 first instance disregarded; it is madein reference to what follows, the ex- 

 planation as to the allotrious points. 



