T h e m a t h e m a t i c s of t h e des ign of a cyl indr ical p r o p e l l a n t g ra in is d iscussed in genera l . Only t h e di rect r ix of t h e cyl indr ical b u r n i n g surface ( the in t e r sec t ion of t h e cyl inder w i t h a p l a n e n o r m a l t o i t s l a t e ra l surface) need be s t ud i ed . D u r i n g b u r n i n g , t h e evolute of t h e di rect r ix is i n v a r i a n t . A cusp convex t o w a r d t h e gas p h a s e r e m a i n s a cusp . A cusp concave toward t h e gas p h a s e becomes a n a rc of a circle. T h e r a t e of increase or decrease of t h e l e n g t h of a n a rc free f rom cusps depends only o n t h e i n c l i n a t i o n of t h e a rc a t i t s e n d p o i n t s . T h e design of a n i n t e rna l l y b u r n i n g p r o p e l l a n t g ra in is d iscussed in m o r e de ta i l , a n d a m e t h o d is descr ibed for des ign ing a g ra in in wh ich t h e b u r n i n g surface a rea is a prescr ibed func t i on of t h e p o r t i o n of t h e web t h i c k n e s s b u r n e d t h r o u g h .